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Theorem map0b 7760
Description: Set exponentiation with an empty base is the empty set, provided the exponent is nonempty. Theorem 96 of [Suppes] p. 89. (Contributed by NM, 10-Dec-2003.) (Revised by Mario Carneiro, 26-Apr-2015.)
Assertion
Ref Expression
map0b (𝐴 ≠ ∅ → (∅ ↑𝑚 𝐴) = ∅)

Proof of Theorem map0b
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 elmapi 7743 . . . 4 (𝑓 ∈ (∅ ↑𝑚 𝐴) → 𝑓:𝐴⟶∅)
2 fdm 5950 . . . . 5 (𝑓:𝐴⟶∅ → dom 𝑓 = 𝐴)
3 frn 5952 . . . . . . 7 (𝑓:𝐴⟶∅ → ran 𝑓 ⊆ ∅)
4 ss0 3925 . . . . . . 7 (ran 𝑓 ⊆ ∅ → ran 𝑓 = ∅)
53, 4syl 17 . . . . . 6 (𝑓:𝐴⟶∅ → ran 𝑓 = ∅)
6 dm0rn0 5250 . . . . . 6 (dom 𝑓 = ∅ ↔ ran 𝑓 = ∅)
75, 6sylibr 222 . . . . 5 (𝑓:𝐴⟶∅ → dom 𝑓 = ∅)
82, 7eqtr3d 2645 . . . 4 (𝑓:𝐴⟶∅ → 𝐴 = ∅)
91, 8syl 17 . . 3 (𝑓 ∈ (∅ ↑𝑚 𝐴) → 𝐴 = ∅)
109necon3ai 2806 . 2 (𝐴 ≠ ∅ → ¬ 𝑓 ∈ (∅ ↑𝑚 𝐴))
1110eq0rdv 3930 1 (𝐴 ≠ ∅ → (∅ ↑𝑚 𝐴) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1474  wcel 1976  wne 2779  wss 3539  c0 3873  dom cdm 5028  ran crn 5029  wf 5786  (class class class)co 6527  𝑚 cmap 7722
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6825
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-fv 5798  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-1st 7037  df-2nd 7038  df-map 7724
This theorem is referenced by:  map0g  7761  mapdom2  7994  ply1plusgfvi  19382
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